Ray EX is the bisector of `angleDEF` and ray EY is the bisector of `angleDEX`. If `angleDEX=42^(@)`, then find `angleYEF` and `angleDEF`.

Answer

To solve the problem step by step, we will use the information given about the angles and their bisectors.

### Step-by-Step Solution:

1. **Identify Given Information:**
- Ray EX is the bisector of angle DEF.
- Ray EY is the bisector of angle DEX.
- Angle DEX = 42°.

2. **Determine Angle DEX:**
- Since ray EY bisects angle DEX, we can find the two equal parts of angle DEX.
- Angle DEX = 42° means:
\[ \text{Angle YEX} = \text{Angle FEX} = \frac{\text{Angle DEX}}{2} = \frac{42°}{2} = 21°. \]

3. **Determine Angle DEF:**
- Since ray EX bisects angle DEF, we can express angle DEF in terms of angle DEX.
- Angle DEF is composed of two equal angles, which are angle DEX and angle FEX.
- Therefore, we have:
\[ \text{Angle DEF} = \text{Angle DEX} + \text{Angle FEX} = 42° + 42° = 84°. \]

4. **Determine Angle YEF:**
- Angle YEF is made up of angle YEX and angle FEX.
- We already know:
- Angle YEX = 21° (from step 2)
- Angle FEX = 42° (since it is equal to angle DEX).
- Therefore:
\[ \text{Angle YEF} = \text{Angle YEX} + \text{Angle FEX} = 21° + 42° = 63°. \]

5. **Final Answers:**
- Angle DEF = 84°.
- Angle YEF = 63°.

### Summary of Results:
- Angle DEF = 84°.
- Angle YEF = 63°.